0580 · 9.6Cumulative frequency diagrams

Building cumulative frequency tables

0:00Imagine recording how long one hundred students take to complete a puzzle. A frequency table tells us how many students finish within each time interval. But we might want the time by which half have finished. A cumulative frequency diagram helps us estimate that time, alongside quartiles, percentiles and the spread of the middle half. The essential idea is a running total. Instead of counting only the students in one interval, we count everyone up to a particular time.

0:25By the end, you will understand why these totals are plotted at upper class boundaries. You will also know how to draw the curve and read information from it. Our first worked example uses the grouped data on the board. Frequency means the number of observations in a class. The class from ten up to and including twenty contains fifteen students. On the board, its notation says that the time is greater than ten and less than or equal to twenty.

0:49That boundary convention matters because every student must belong to exactly one class. Someone taking exactly twenty minutes belongs to the class ending at twenty, not the next class. All the completion times are greater than zero and no greater than sixty minutes. The frequencies add to one hundred, which is our total number of observations. Cumulative frequency means the total frequency up to and including a particular value. Think of moving the finishing deadline later.

1:13At ten minutes, we count the first group. At twenty minutes, we keep those students and add the next group. The total accumulates rather than starting again for each interval. For the first boundary, the cumulative frequency is ten. Adding fifteen gives twenty-five students who finish within twenty minutes. The next class adds twenty-five, making fifty by thirty minutes. Another twenty-five brings the total to seventy-five by forty minutes. We then add fifteen to seventy-five, obtaining ninety by fifty minutes.

1:42The final ten brings the cumulative frequency to one hundred at sixty minutes. This matches the total frequency, so it provides a useful check. Every observation must have been counted by the final upper boundary. The completed table answers questions about everyone below a boundary. A cumulative frequency of seventy-five at forty minutes means seventy-five students finish in forty minutes or less. It does not mean seventy-five students belong to the class from thirty up to and including forty.

2:08That individual class contains only twenty-five students. You can recover an individual class frequency by subtracting neighbouring cumulative totals. For the class from forty up to and including fifty, subtract seventy-five from ninety. The result is fifteen students. Subtraction removes everyone already counted by forty minutes, leaving only the additional students in the next interval.

Drawing cumulative frequency curves

2:30To draw the diagram, put completion time on the horizontal axis and cumulative frequency on the vertical axis. Choose scales that use a substantial part of the grid. Label the measured quantity and its unit. The cumulative frequency axis records a count, so its maximum must accommodate all one hundred students. Each plotted point pairs an upper class boundary with its cumulative frequency. At twenty minutes, the running total is twenty-five, so plot the point with horizontal coordinate twenty and vertical coordinate twenty-five.

2:59We use twenty because that total includes the entire class ending there. A class midpoint would represent the wrong threshold. Plot all the points clearly as small crosses. Include the starting point with time zero and cumulative frequency zero, because no completion times are at or below zero. Join the points with a smooth curve, as required for this diagram. Do not replace it with a bar chart or a jagged series of straight segments.

3:22The curve cannot decrease as we move from left to right. Allowing more time cannot remove students who have already finished. A steeper section means more observations lie within that stretch of the horizontal axis. A flatter section means fewer observations are being added over the same width. At the class boundaries, our cumulative totals come directly from the table. Between those boundaries, the grouped data do not tell us the exact individual completion times.

3:47The smooth curve lets us estimate the totals between known points. This is why answers read from the curve are estimates rather than exact individual measurements.

Finding medians and quartiles

3:56Our second worked example asks us to estimate the median, quartiles and interquartile range. Imagine arranging all the students from the shortest completion time to the longest. The median divides this ordered group into two halves. On a cumulative frequency diagram, we therefore find the level representing half the total frequency. Divide one hundred by two to get a cumulative frequency of fifty. Start at fifty on the vertical axis and move horizontally to the curve.

4:22Then move vertically down to the time axis, where the reading is thirty minutes. The estimated median is therefore thirty minutes, with approximately half the students finishing by then. Notice the difference between the position and the value. Fifty tells us how far through the group we need to count. Thirty minutes is the completion time at that position. Reporting fifty as the median would confuse a number of students with a time.

4:44The lower quartile marks twenty-five per cent of the way through the ordered data. Divide one hundred by four to obtain a cumulative frequency of twenty-five. Reading across to the curve and down gives twenty minutes. Approximately twenty-five per cent of students therefore finish in twenty minutes or less. The upper quartile marks seventy-five per cent of the way through the ordered data. Multiply one hundred by three, then divide by four, giving seventy-five.

5:09At that cumulative frequency, the curve gives forty minutes. Approximately seventy-five per cent finish by forty minutes, leaving about twenty-five per cent who take longer. The interquartile range is the upper quartile minus the lower quartile. Subtract twenty from forty to obtain twenty minutes. This describes the width of the interval containing the middle fifty per cent of completion times. It measures spread, rather than telling us the typical completion time.

5:34Our readings landed on plotted points, but that will not always happen. Usually, a quartile or median level meets the curve somewhere between the marked crosses. Follow the same horizontal then vertical route. Read the curve itself rather than choosing the nearest point or the nearest class boundary. For cumulative frequency diagrams, use half the total for the median and the appropriate fractions for the quartiles. Do not switch to a position formula used for a list of individual values.

6:00Here, we are locating proportions of the whole group on a continuous curve. The original individual observations are not available. Percentiles extend the same idea beyond the quartiles. The sixtieth percentile is the value at or below which approximately sixty per cent of observations lie. The median is the fiftieth percentile. The lower and upper quartiles are the twenty-fifth and seventy-fifth percentiles respectively, so these are all readings from the same curve.

Percentiles and interval counts

6:25Our third worked example asks for the sixtieth percentile and the number of students taking longer than that time. It also asks how many finish after twenty minutes but within that percentile time. First, translate the required percentage into a cumulative frequency. Then use the curve to find the corresponding completion time. Calculate 60 ÷ 100, multiplied by the total frequency of one hundred. This gives a cumulative frequency of sixty.

6:51Move horizontally from sixty to the curve, then vertically down to the time axis. The estimated sixtieth percentile is thirty-four minutes, meaning approximately sixty per cent finish within that time. To estimate the number taking longer than thirty-four minutes, subtract sixty from the total of one hundred. The result is forty students. The curve counts those who finish by the threshold, not those above it. Subtracting from the total gives the remaining group.

7:16For completion times greater than twenty but no greater than thirty-four, subtract the cumulative total at the lower threshold from the upper one. 60 − 25 gives thirty-five students. To express that as a percentage, divide thirty-five by one hundred and multiply by one hundred. The estimated percentage is thirty-five per cent. This subtraction works because both cumulative totals contain everyone who finished within twenty minutes. Removing the smaller total cancels those earlier observations.

7:43Only the students between the two thresholds remain. Read carefully whether a question asks for a time, a number of observations or a percentage. Our next worked example reverses the direction of reading. It asks for an estimate of how many students take more than forty-five minutes. This time, start at forty-five on the horizontal time axis. Move vertically up to the curve, then horizontally across to the cumulative frequency axis.

8:08The reading is approximately eighty-three. So about eighty-three students finish within forty-five minutes. Subtracting eighty-three from the total of one hundred leaves seventeen students. Because the total is one hundred, that group is also seventeen per cent of the class. A reading such as eighty-three is an estimate from a hand-drawn curve. Mark schemes usually accept a small range of values around the correct reading. Your construction lines show the examiner exactly where your answer came from.

8:34Without them, a slightly different reading may earn no method mark.

Comparing groups and avoiding errors

8:38Interpreting cumulative frequency diagrams often means comparing two groups. Suppose a second group of one hundred students attempts the same puzzle. Its curve gives a median of twenty-six minutes and an interquartile range of twelve minutes. Our first group had a median of thirty minutes and an interquartile range of twenty minutes. A strong comparison makes one point about the average and one about the spread, both in context. The second group has the lower median, so on average its students finished faster.

9:04It also has the smaller interquartile range. Its middle fifty per cent of completion times lie closer together, so they are more consistent. In an examination, you may be asked to complete a cumulative frequency table, draw the diagram, or estimate values from it. Show the running totals when completing the table. On the graph, mark the points clearly and draw the smooth curve. For readings, construction lines make your method visible and reduce mistakes.

9:29These methods must work without a calculator on Paper two. Paper four allows a calculator, but the plotting and reading process stays the same. Calculate percentage positions accurately and avoid rounding during the working. For graph estimates, use precision supported by the scale rather than automatically giving three significant figures. Common mistakes include plotting ordinary frequencies or using class midpoints instead of upper boundaries. These choices do not show the total up to a threshold.

9:56Another error is starting a percentile reading on the horizontal axis instead of the cumulative frequency axis. Begin with the required count, then find its corresponding data value. A cumulative frequency diagram turns grouped frequencies into a picture of how observations accumulate. Read from a cumulative count to find a median, quartile or percentile. Read from a data value to estimate how many observations lie at or below it. Subtract cumulative totals for an interval, and subtract quartiles to find the interquartile range.